Logical connections in the many-sorted setting

Transcription

1 Logical connections in the many-sorted setting Jiří Velebil Czech Technical University in Prague Czech Republic joint work with Alexander Kurz University of Leicester United Kingdom AK & JV AsubL4 1/24

2 Plan of the talk 1 Classical modal logic (syntax, semantics, Jónsson-Tarski, Goldblatt-Thomason theorems) can be recovered from a logical conection. 2 A general nature of a logical connection. 3 Results: syntax, semantics, Jónsson-Tarski and Goldblatt-Thomason theorems. 4 More examples: what we can choose for propositional part of a logic. AK & JV AsubL4 2/24

3 (Bonsangue & Kurz, 2004) Classical modal logic can be retrieved from the data 1 models will have sets of states = choice of the category Set of sets and mappings 2 Kripke frames = coalgebras for P : Set Set, one such is c : X P(X ) 3 choice of the propositional part of the logic = the variety BA of Boolean algebras 4 an adjunction Set op Stone Pred BA X Stone(A) A Pred(X ) of a special kind: it is given by an object Ω = 2 living in both Set and BA (schizophrenic object) = logical connection AK & JV AsubL4 3/24

4 The adjunction Stone Pred in more detail The assignment X Set(X, 2) has a canonical lift to Boolean algebras Set op Pred BA X Set(X,2) U Set Intution: Pred gives truth distributions. The assignment A BA(A, 2) has a canonical lift to sets Intuition: Stone gives theories. BA op Stone Set A BA(A,2) Set V =Id AK & JV AsubL4 4/24

5 The syntax of the modal part Given P : Set Set, define L : BA BA as the composite a BA Stone Set op P op Set op Pred BA Algebras for L = modal algebras (= BAOs), one such is α : LA A. α computes flat terms: (, a 0,..., a n 1 ) LA (a 0,..., a n 1 ) A a A tiny technicality here, if we want a finitary language. AK & JV AsubL4 5/24

6 Modalities We obtain all finitary modalities, one such is a map i.e., it is a truth table : P(2 n ) 2 0,..., n 1 r rows r = subsets of 2 n, values v = elements of Ω = 2 Thus: the modal language of predicate liftings (Kupke, Kurz, Schröder,... ). v AK & JV AsubL4 6/24

7 Modal algebras = BAOs = algebras for L Denote their category by BA L. There is a chain of adjunctions BA L F L F BA Set U L U F L F (At) F (At) At (BAO of modal formulas) (Boolean algebra of prop. formulas) (set of atomic propositions) that is finitary and monadic = BAOs form a variety. AK & JV AsubL4 7/24

8 Semantic of the logic There is, canonically, a commutative square (Set P ) op Pred BA L (V P ) op Set op Pred BA U L allowing for val c : F L F (At) Pred (X, c), ϕ {x X x val c ϕ} from a given valuation val : At UU L Pred (X, c), p {x X p holds in x} for each coalgebra c : X PX. AK & JV AsubL4 8/24

9 Semantic map = homomorphism By construction: val c : F L F (At) Pred (X, c) is a homomorphism of modal algebras, i.e.,.(ϕ 0,..., ϕ n 1 ) val c = [[ ]] Pred (X,c) ( ϕ 0 val c,..., ϕ n 1 val c ) holds. AK & JV AsubL4 9/24

10 Unravelling the definition of c in c : X P(X ) To see whether holds: x c.(ϕ 0,..., ϕ n 1 ) 1 In each x X determine, whether x c ϕ j, j n. This gives a map X 2 n, hence a map P(X ) P(2 n ). 2 Compute the composite X c P(X ) P(2 n ) 2 and evaluate at x. Hence: the classical Kripke semantics. AK & JV AsubL4 10/24

11 Further results on classical modal logic 1 Jónsson-Tarski completeness theorem: the unit η of Stone Pred is a (regular) mono and it lifts to BA L (Kurz & Rosický, 2006) (Set P ) op (V P ) op Set op Stone Pred Stone Pred BA L BA U L η A : (A, a) Pred Stone (A, a) η A : A PredStone(A) AK & JV AsubL4 11/24

12 Further results on classical modal logic 2 Goldblatt-Thomason theorem on modal definability: the counit ε of Stone Pred provides us with the notion of ultrafilter extension ε (X,c) : Stone Pred (X, c) (X, c) in (Set P ) op or, by passing from (Set P ) op to Set P, (ε ) op (X,c) : (X, c) (Stone ) op (Pred ) op (X, c) in Set P (Kurz & Rosický, 2007) AK & JV AsubL4 12/24

13 What has been studied instead of P : Set Set 1 Kripke-polynomial functors on Set (Kupke, Kurz, Moss, Pattinson, Schröder, Venema,... ): T ::= Id const X T T T + T PT essentially: modal logics of automata. 2 measure-polynomial functors (Goldblatt, Moss, Viglizzo,... ): T ::= Id const X T T T + T T Hence: modal logic on the category Meas of measurable spaces. AK & JV AsubL4 13/24

14 The main idea Replace Set V =Id by a general situation Spa V BA F Set Ω = 2 Set Set U Alg [S, Set] Ω : S A Set [A, Set] with Ω schizophrenic, i.e., in both Spa and Alg. F U AK & JV AsubL4 14/24

15 Examples of Alg = propositional part of the logic 1 Any finitary variety: distributive lattices, Heyting algebras, residuated lattices. 2 A =finite ordinals and all maps, H-algebras for HX = X X + X + δx, (δx )(n) = X (n + 1) H-algebra = 1st order formulas 3 A =ingredients of a Kripke-polynomial T : Set Set. Alg = [A, BA]. This gives many-sorted modal logic of Rössiger for description of automata. AK & JV AsubL4 15/24

16 Examples of Alg, cont. 4 H-monoids for a strong H : [A, Set] [A, Set], where A is promonoidal. Example: A =finite ordinals and all maps, syntax of λ-calculus with α-conversion (Fiore, Plotkin & Turi, 1999), HX = X X + δx. H-monoid = presheaf of λ-terms. More generally: a syntax given by variable binding (Tanaka & Power, 2000). Examples include: linear λ-calculus, logic of bunched implications,... AK & JV AsubL4 16/24

17 What schizophrenic modules are Ω : S A Set such that 1 Ω lives in both Spa and Alg: A Ω Spa Spa S Ω Alg a Ω(,a) V Alg s Ω(s, ) [S, Set] [A, Set] 2 Roughly: for every a and s, there are canonical Spa- and Alg-structures on U {A, Ω(s, )} {X, Ω(, a)} for every algebra A and space X. AK & JV AsubL4 17/24

18 Results (Kurz & JV, 2010) 1 Schizophrenic modules Ω give rise to logical connections Stone Pred : Spa op Alg One can even work enriched, i.e., with V -categories (replace Set by a symmetric monoidal closed V ). Technique: a initial lifts along U : Alg [A, Set] and V : Spa [S, Set]. 2 Given T : Spa Spa, one can define L : Alg Alg, giving rise to 1 Modal algebras Alg L forming a variety over [A, Set], if Alg was a variety over [A, Set]. (Variety = the adjunction F U is finitary and monadic.) 2 Syntax and semantics can be defined. The rôle of Ω = external truth values. a Generalizing Porst & Tholen, 1991 AK & JV AsubL4 18/24

19 Intuition behind Pred and Stone There is a canonical lift Spa op Pred Alg X {X,Ω } U [A, Set] Here, elements of {X, Ω } : A Set are truth-distributions on X : S Set, one such is a a natural transf. from X to Ω(, a), naturally in a Similarly for Stone, elements of Stone are ultrafilters (= theories ). AK & JV AsubL4 19/24

20 Examples of logical connections 1 Ω=any distributive/residuated lattice, it gives rise to Stone Pred : Set op DL, Stone Pred : Set op RL 2 Spa=measurable spaces, Alg=Boolean algebras, Ω = 2. The resulting logic is the logic of Harsanyi spaces (Moss & Viglizzo, 2004). Modality = a measurable map : T (2 n ) 2 i.e., a measurable subset of T (2 n ). AK & JV AsubL4 20/24

21 More examples of logical connections, enriched 4 V =posets, Ω=two-element chain, it gives rise to Stone Pred : Pos op DL The resulting logic is expressive (Kapulkin, Kurz & JV, 2010) for good T : Pos Pos. 5 V =2, V -categories are preorders. Any Ω : S A V is a (monotone) binary relation and it is schizophrenic. Logical connection = Galois connection between upper sets of S and A. 6 Another facet of the previous: V =Abelian groups, A, S rings with unit, Ω= any A -S -bimodule. The resulting logical connection is the one known from module theory. AK & JV AsubL4 21/24

22 General form of modalities 1 Arities are finitely presentable objects of [A, Set]. 2 For an f.p. n : A Set, an n-ary modality is : a (a) : T {n, Ω} Ω(, a), functorial in a and every (a) is natural in s, since Spa lives over [S, Set]. 3 n-ary can be applied only to an n-ad in A, i.e., to n UA allowing for partially defined modalities. AK & JV AsubL4 22/24

23 More results (Nentvich, Petrişan & JV, 2010) Given T : Spa Spa, the existence of (Spa T ) op (V P ) op Spa op Stone Pred Stone Pred Alg L Alg U L allows to formulate and prove 1 Jónsson-Tarski theorem 2 Goldblatt-Thomason theorem Some side conditions on Stone Pred are needed, however. AK & JV AsubL4 23/24

24 References H.-E. Porst and W. Tholen, Concrete dualities, in: Category theory at work (H. Herrlich, H.-E. Porst, eds.), Heldermann Verlag, Berlin 1991, M. Bonsangue and A. Kurz, Duality for logics of transition systems, FOSSACS 2005 A. Kurz and J. Rosický, Strongly complete logics for coalgebras, preprint 2006 A. Kurz and J. Rosický, The Goldblatt-Thomason theorem for coalgebras, CALCO 2007 K. Kapulkin, A. Kurz and J. V., Expressivity of coalgebraic logic over posets, CMCS 2010, A. Kurz and J. V., Enriched logical connections, preprint 2010 AK & JV AsubL4 24/24